Whakaoti i te {l}{y=x/20}{y=(80+x)*1/30}

$$\left. \begin{array} { l } { y = \frac { x } { 20 } } \\ { y = ( 80 + x ) \times \frac { 1 } { 30 } } \end{array} \right.$$

Whakaoti mō y, x

$$x=160$$

$$y=8$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te \frac{x}{20} mai i ngā taha e rua.

$$y-\frac{x}{20}=0$$

Whakareatia ngā taha e rua o te whārite ki te 20.

$$20y-x=0$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 80+x ki te \frac{1}{30}.

$$y=\frac{8}{3}+\frac{1}{30}x$$

Tangohia te \frac{1}{30}x mai i ngā taha e rua.

$$y-\frac{1}{30}x=\frac{8}{3}$$

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$20y-x=0,y-\frac{1}{30}x=\frac{8}{3}$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te y mā te wehe i te y i te taha mauī o te tohu ōrite.

$$20y-x=0$$

Me tāpiri x ki ngā taha e rua o te whārite.

$$20y=x$$

Whakawehea ngā taha e rua ki te 20.

$$y=\frac{1}{20}x$$

Whakakapia te \frac{x}{20} mō te y ki tērā atu whārite, y-\frac{1}{30}x=\frac{8}{3}.

$$\frac{1}{20}x-\frac{1}{30}x=\frac{8}{3}$$

Tāpiri \frac{x}{20} ki te -\frac{x}{30}.

$$\frac{1}{60}x=\frac{8}{3}$$

Me whakarea ngā taha e rua ki te 60.

$$x=160$$

Whakaurua te 160 mō x ki y=\frac{1}{20}x. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=\frac{1}{20}\times 160$$

Whakareatia \frac{1}{20} ki te 160.

$$y=8$$

Kua oti te pūnaha te whakatau.

$$y=8,x=160$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te \frac{x}{20} mai i ngā taha e rua.

$$y-\frac{x}{20}=0$$

Whakareatia ngā taha e rua o te whārite ki te 20.

$$20y-x=0$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 80+x ki te \frac{1}{30}.

$$y=\frac{8}{3}+\frac{1}{30}x$$

Tangohia te \frac{1}{30}x mai i ngā taha e rua.

$$y-\frac{1}{30}x=\frac{8}{3}$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$20y-x=0,y-\frac{1}{30}x=\frac{8}{3}$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}20&-1\\1&-\frac{1}{30}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\\frac{8}{3}\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}20&-1\\1&-\frac{1}{30}\end{matrix}\right).

$$inverse(\left(\begin{matrix}20&-1\\1&-\frac{1}{30}\end{matrix}\right))\left(\begin{matrix}20&-1\\1&-\frac{1}{30}\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}20&-1\\1&-\frac{1}{30}\end{matrix}\right))\left(\begin{matrix}0\\\frac{8}{3}\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}20&-1\\1&-\frac{1}{30}\end{matrix}\right))\left(\begin{matrix}0\\\frac{8}{3}\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}20&-1\\1&-\frac{1}{30}\end{matrix}\right))\left(\begin{matrix}0\\\frac{8}{3}\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right) te poukapa kōaro, nō reira ka taea te tuhi anō te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{\frac{1}{30}}{20\left(-\frac{1}{30}\right)-\left(-1\right)}&-\frac{-1}{20\left(-\frac{1}{30}\right)-\left(-1\right)}\\-\frac{1}{20\left(-\frac{1}{30}\right)-\left(-1\right)}&\frac{20}{20\left(-\frac{1}{30}\right)-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}0\\\frac{8}{3}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{10}&3\\-3&60\end{matrix}\right)\left(\begin{matrix}0\\\frac{8}{3}\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}3\times \frac{8}{3}\\60\times \frac{8}{3}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}8\\160\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=8,x=160$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te \frac{x}{20} mai i ngā taha e rua.

$$y-\frac{x}{20}=0$$

Whakareatia ngā taha e rua o te whārite ki te 20.

$$20y-x=0$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 80+x ki te \frac{1}{30}.

$$y=\frac{8}{3}+\frac{1}{30}x$$

Tangohia te \frac{1}{30}x mai i ngā taha e rua.

$$y-\frac{1}{30}x=\frac{8}{3}$$

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$20y-x=0,y-\frac{1}{30}x=\frac{8}{3}$$

Kia ōrite ai a 20y me y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 20.

$$20y-x=0,20y+20\left(-\frac{1}{30}\right)x=20\times \frac{8}{3}$$

Whakarūnātia.

$$20y-x=0,20y-\frac{2}{3}x=\frac{160}{3}$$

Me tango 20y-\frac{2}{3}x=\frac{160}{3} mai i 20y-x=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$20y-20y-x+\frac{2}{3}x=-\frac{160}{3}$$

Tāpiri 20y ki te -20y. Ka whakakore atu ngā kupu 20y me -20y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-x+\frac{2}{3}x=-\frac{160}{3}$$

Tāpiri -x ki te \frac{2x}{3}.

$$-\frac{1}{3}x=-\frac{160}{3}$$

Me whakarea ngā taha e rua ki te -3.

$$x=160$$

Whakaurua te 160 mō x ki y-\frac{1}{30}x=\frac{8}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y-\frac{1}{30}\times 160=\frac{8}{3}$$

Whakareatia -\frac{1}{30} ki te 160.

$$y-\frac{16}{3}=\frac{8}{3}$$

Me tāpiri \frac{16}{3} ki ngā taha e rua o te whārite.

$$y=8$$

Kua oti te pūnaha te whakatau.

$$y=8,x=160$$